HYACINTHOS 29515

[Tran Quang Hung]:

 

Let ABC be a triangle.
A'B'C' is cevian triangle of circumcenter O.
Perpendicular bisectors of AA', BB', CC' bound triangle A"B"C".
Then O lies on Euler line of A"B"C".
Which is point O of A"B"C"?

[Angel Montesdeoca]:

**** The point O of A"B"C" is the is midpoint of X(26) and X(23709), with first barycentric coordinate:

a^2 (a^2 (b^2+c^2)-(b^2-c^2)^2) (a^10-4 a^8 (b^2+c^2)+a^6 (6 b^4+8 b^2 c^2+6 c^4)
-a^4 (4 b^6+3 b^4 c^2+3 b^2 c^4+4 c^6)+a^2 (b^8-2 b^6 c^2-b^4 c^4-2 b^2 c^6+c^8)
+b^2 c^2 (b^2-c^2)^2 (b^2+c^2)).

Lies on lines X(i)X(j) for these {i, j}:
{2,13467}, {3,54}, {26,23709}, {30,31867}, {186,14978}, {1658,32428}, {2070,32551}, {6368,14809}, {7502,25043}, {7512,14652}, {11264,23195}.

 Angel Montesdeoca

[Antreas P. Hatzipolakis]:

Dear Angel

This is a new triangle center and will be included in ETC, but Tran meant to ask:

Which point is the circumcenter O of ABC wrt triangle A"B"C" (lying on the Euler line of A"B"C")

Antreas P. Hatzipolakis